Signals and Systems (GPS and Galileo Receiver) Part 1

The concepts of a signal and a system are crucial to the topic. We will consider time as well as frequency domain models of the signals. We focus on signals and system components that are important to study software-defined GPS and Galileo receiver design.

Characterization of Signals

In satellite positioning systems, we encounter two classes of signals referred to as deterministic and random signals. Deterministic signals are modeled by explicit mathematical expressions. The signalstmp2D1_thumbare examples of deterministic signals. A random signal, on the other hand, is a signal about which there is some degree of uncertainty. An example of a random signal is a received GPS signal: the received signal contains beside the information bearing signal also noise from disturbances in the atmosphere and noise from the internal circuitry of the GPS receiver.

Now some basic topics on deterministic and stochastic signal theory are reviewed and simultaneously we establish a notation.

A reader familiar with random processes knows concepts like autocorrelation function, power spectral density function (or power spectrum), and bandwidth. These concepts can be applied for deterministic signals as well, and that is exactly what we intend to do in the following.

Continuous-Time Deterministic Signals

Let us consider a deterministic continuous-time signal x(t), real- or complex-valued with finite energy defined astmp2D3_thumbThe symbol denotes the absolute value, or magnitude, of the complex quantity. In the frequency domain this signal is represented by its Fourier transform:

tmp2D5_thumb

wheretmp2D6_thumband the variabletmp2D7_thumbdenotes angular frequency. By definitiontmp2D8_thumband the units fortmp2D9_thumband f are radian and cycle, respectively. In general, the Fourier transform is complex:

tmp2D14_thumb

The quantitytmp2D15_thumbis often referred to as the spectrum of the signal x (t) because the Fourier transform measures the frequency content, or spectrum, of x(t). Similarly, we refer totmp2D16_thumbas the magnitude spectrum of x (t), and to argtmp2D17_thumb= arctantmp2D18_thumbas the phase spectrum of x (t). Moreover, we refer to tmp2D19_thumbas the energy density spectrum of x (t) because it represents the distribution of signal energy as a function of frequency. It is denotedtmp2D20_thumb The inverse Fourier transform x(t) oftmp2D21_thumbis

tmp2D29_thumb

We say that x (t) andtmp2D30_thumbconstitute a Fourier transform pair:

tmp2D32_thumb

The energy density spectrumtmp2D33_thumbof a deterministic continuous-time signal x (t) can also be found by means of the (time-average) autocorrelation function (ACF) of the finite energy signal x (t). Let * denote complex conjugation, and then the ACF of x (t) is defined as

tmp2D35_thumb

is defined as and the energy density spectrumtmp2D36_thumb is defined as

tmp2D37_thumb

Again, we say thattmp2D38_thumbconstitute a Fourier transform pair:

tmp2D40_thumb

Discrete-Time Deterministic Signals

Let us suppose that x(n) is a real- or complex-valued deterministic sequence, where n takes integer values, and which is obtained by uniformly sampling the continuous-time signal x(t); read Section 1.2. If x(n) has finite energytmp2D41_thumb

tmp2D42_thumbthen it has the frequency domain representation (discrete-time Fourier transform)

tmp2D45_thumb

or equivalently

tmp2D46_thumb

It should be noted that X (f) is periodic with a period of one andtmp2D47_thumbis periodic with a period oftmp2D48_thumb

The inverse discrete-time Fourier transform that yields the deterministic sequence x(n) fromtmp2D49_thumbor X (f) is given by

tmp2D53_thumb

Notice that the integration limits are related to the periodicity of the spectra. We refer totmp2D54_thumbas the energy density spectrum of x(n) and denote it as

tmp2D57_thumb

The energy density spectrumtmp2D58_thumbof a deterministic discrete-time signal x(n) can also be found by means of the autocorrelation sequence

tmp2D60_thumb

via the discrete-time Fourier transform

tmp2D62_thumb

That is, for a discrete-time signal, the Fourier transform pair is

tmp2D63_thumb

Unit Impulse

In signal analysis a frequently used deterministic signal is the unit impulse. In continuous time the unit impulsetmp2D64_thumb, also called the delta function, may be defined by the following relation:

tmp2D66_thumb

where x (t) is an arbitrary signal continuous at t = 0. Its area istmp2D67_thumbRectangular pulse.

FIGURE 1.1. Rectangular pulse.

In discrete time the unit sample, also called a unit impulse sequence, is defined as

tmp2D69_thumb

It follows that a continuous-time signal x (t) may be represented as

tmp2D70_thumb

Similarly, a sequence x(n) may be represented as

tmp2D71_thumb

The Fourier transform of the unit impulsetmp2D72_thumbis given by

tmp2D74_thumb

which gives us the following Fourier transform pair:

tmp2D75_thumb

The spectrum of the unit sample is obtained by

tmp2D76_thumb

which gives us the following Fourier transform pair:

tmp2D77_thumb

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